In the case of a sphere, solution to Problem REF  is known on the unit circle \(S^1\)  for every \(N\ge 1\)  and \(f(t)=t^{-s/2}\) , \(s>0\) , and \(f(t)=-t^{-s/2}\) , \(-1\le s<0\) , which correspond to the Riesz potential, as well as for \(f(t)=\frac{1}{2}\ln \frac{1}{t}\) , which corresponds to the logarithmic potential, see the works by Stolarsky, Ambrus, Nikolov, Rafailov, Ball, Erdélyi, Saff, Hardin, and Kendall [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. The solution is the set of vertices of a regular \(N\) -gon inscribed in \(S^1\) .
In fact, paper [6]} established this for arbitrary interactions given by a decreasing and convex function of the geodesic distance on \(S^1\) . Optimal \(N\) -point configurations for polarization on \(S^1\)  were also characterized by Bosuwan and Ruengrot [8]} for the Riesz potential with \(s=-2,-4,\ldots ,2-2N\) , \(N\ge 2\) .
